Finite Element Modeling of Biomolecular Systems in Ionic Solution

Abstract

An accurate finite element method is introduced to solve the two most commonly used continuum models in computational biophysics: Poisson–Boltzmann (PB) equation and Poisson–Nernst–Planck (PNP) equations. They describe equilibrium and non-equilibrium (with diffusion existed) properties of ionic liquid, respectively. Both models involve two domains (solvent and solute) with distributed singular permanent charges inside biomolecules (solute domain) and a dielectric jump at the interface between solvent and solute. A stable regularization scheme is described to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and regular, well-posed PB/PNP equations are formulated. The interface conditions for electric potential are also explicitly enforced to be satisfied. An inexact-Newton method is used to solve the nonlinear elliptic PB equation and the coupled steady-state PNP equations; while an Adams–Bashforth–Crank–Nicolson method is devised for time integration for the unsteady electrodiffusion. The numerical methods are shown to be accurate and stable by various tests of real biomolecular electrostatic and diffusion problems.